MATHEMATICS OF OPERATIONS RESEARCH
Vol. 33, No. 2, May 2008, pp. 375–402
issn 0364-765X eissn 1526-5471 08 3302 0375
informs
®
doi 10.1287/moor.1070.0298
© 2008 INFORMS
Fluid Limits for Processor-Sharing Queues with Impatience
H. Christian Gromoll
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Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903, USA,
gromoll@virginia.edu, http://www.faculty.virginia.edu/gromoll/
Philippe Robert
INRIA-Rocquencourt, RAP Project, Domaine de Voluceau, BP 105, 78153 Le Chesnay, France,
Philippe.Robert@inria.fr, http://www-rocq.inria.fr/~robert
Bert Zwart
Georgia Institute of Technology, Stewart School of ISyE, 765 Ferst Drive, Atlanta, Georgia 30332, USA,
bertzwart@gatech.edu
We investigate a processor-sharing queue with renewal arrivals and generally distributed service times. Impatient jobs may
abandon the queue or renege before completing service. The random time representing a job’s patience has a general distribution and may be dependent on its initial service time requirement. A scaling procedure that gives rise to a fluid model with
nontrivial yet tractable steady state behavior is presented. This fluid model captures many essential features of the underlying
stochastic model, and it is used to analyze the impact of impatience in processor-sharing queues.
Key words: processor sharing; queues with impatience; measure-valued process; fluid limits; delay-differential equations;
empirical processes
MSC2000 subject classification: Primary: 60K25, 60K30, 60G57, 60F17; secondary: 90B15, 90B22
OR/MS subject classification: Primary: queues, limit theorems; secondary: scheduling
History: Received April 5, 2006; revised April 30, 2007, and August 30, 2007.
1. Introduction.
Processor-sharing policy and impatience. Processor-sharing (PS) policies were originally proposed as
models of time sharing in computer operating systems. Recently, generalizations of this discipline have been
used to describe data transfers in congested routes through the Internet (see Roberts and Massoulié [25] and
Kelly and Williams [18] and references therein). This has created considerable renewed interest in the analysis
of PS policies.
This paper studies the behavior of a GI/GI/1 queue serving impatient jobs according to the PS policy: If
there are N jobs in the queue, each job receives simultaneous service at rate 1/N . An impatient job has a random
initial lead time in addition to its service time. Such a job has a deadline equal to its arrival time plus its initial
lead time; if the job has not completed service when the deadline expires, it abandons the queue (or reneges)
and therefore does not complete service. For example, the timeout of a transmission control protocol (TCP) flow
through the Internet can be thought of as the expiration of a random deadline and subsequent reneging of the
flow.
The impact of impatience on PS queues is greater than for first-in-first-out (FIFO) queues. A typical job
that abandons a FIFO queue will do so while waiting to begin service. In contrast, a job that abandons a PS
queue will have already received partial service. Because this partial service is wasted, impatience may create
significant overhead for a PS server.
There is a large literature on queueing models with impatience under the FIFO discipline. An early paper
by Barrer [1] considers an example arising in a military application. Stanford [27] surveys the literature in
this domain (see also Stanford [26] and Boots and Tijms [5]). This body of work focuses primarily on exact
performance analysis. Ward and Glynn [29] have recently obtained a diffusion approximation for single channel
queues. There are also various studies of multiserver queues with abandonments, motivated by call center
applications; see the survey by Bonald and Massoulié [3], Gans et al. [10], and references therein.
There is some related literature treating other policies, but in the context of soft deadlines. Jobs with soft
deadlines are not impatient; they remain in the system until completing service, even if their deadlines have
expired. In particular, these queues are work conserving in the sense that the server must fully process all work
arriving to the system. Results for such models describe the extent to which overdue jobs are produced by the
underlying service discipline, without the effect of abandonments. Doytchinov et al. [9] and Kruk et al. [20, 21]
investigate the heavy traffic behavior of various systems using the earliest deadline first and FIFO policies.
Gromoll and Kruk [12] describe the heavy traffic behavior of a PS queue incorporating a fairly general structure
of soft deadlines.
375
376
Gromoll, Robert, and Zwart: Fluid Limits for Processor-Sharing Queues with Impatience
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For PS queues with impatience, only a few results are known. Coffman et al. [7] cover the special case
of exponential service times and lead times, where the lead time and service time are independent. Guillemin
et al. [14] consider heavy tailed service times, and obtain some results on the reneging behavior of large jobs by
analyzing the tail behavior of the sojourn time distribution. Using some approximations, Bonald and Roberts [4]
analyze the steady state of a system with general service times and some dependence between service times and
lead times.
Results of the paper. This paper analyzes the PS queue with impatience using fluid limits. The dynamics
of the system are represented as a measure-valued process: the system state at time t ≥ 0 is represented as a
random point measure t on 0 × 0 , such that t has a point mass at b d ∈ 0 × 0 if
and only if there is a job in the system at time t with residual service time b and residual lead time d. See
Jean-Marie and Robert [16] and Doytchinov et al. [9] for similar representations of residual service times in
single server queues. This setup enables a fairly general analysis. The case of a general joint distribution of
service times and initial lead times with possible dependence of the two random variables is included in our
setting.
Under mild assumptions, we show that under a convenient scaling, a family of measure-valued processes
associated with t is tight and converges in distribution to a limit t. For t ≥ 0, t is a nonnegative
measure on 0 × 0 . This fluid limit is characterized as the solution of a functional Equation (2.9), which
can be viewed as a time-changed functional differential equation.
The overloaded case > 1, which is our main focus, presents a nontrivial and interesting steady state behavior.
The total fluid mass in the system at equilibrium (the fluid analogue of the total number of jobs) is shown to
be the solution z of a simple fixed point Equation (3.4). Moreover, the steady state of the fluid model, that is,
the limit of t as t goes to infinity, is a measure on 0 × 0 , which has a simple expression (2.15) in
terms of z .
These results provide significant insight into the qualitative properties of PS queues with impatience. An
interpretation of the fixed point Equation (3.4) is given and used to analyze the total number of jobs in the
system, and to estimate the long run fraction of jobs that renege. The impact of the variability of service times
and lead times, as well as other properties of this model, are investigated in Gromoll et al. [11].
In contrast to previous work on PS models, the server considered here might only process a fraction of the
service requirement of a job. This creates an important difference: The workload process does not coincide with
that of a FIFO queue, a fact that previous work has exploited heavily. For this reason, analysis of the fluid model
is more intricate. A different approach to prove existence, uniqueness, and convergence to steady state of fluid
model solutions is used. It is shown that there exists a maximal fluid model solution and, using monotonicity
arguments, the properties of fluid limits are investigated under fairly general assumptions.
Organization. A detailed description of the model and main results are presented in §2. Qualitative properties of the fluid model are analyzed in §3. Section 4 is devoted to examples. Sections 5 and 6 are concerned
with convergence to the fluid limit. Section 5 establishes tightness, and §6 characterizes limit points.
2. Model description and results. This section gives a detailed description of the stochastic processes
associated to this queue, as well as a summary of our main results.
2.1. Stochastic model. The stochastic model consists of the following: a processor-sharing server working
at unit rate from an infinite capacity buffer, a collection of stochastic primitives E·, Bi Di describing, respectively, the arrival process, service requirements, and initial lead times of jobs, and a random initial condition
specifying the state of the system at time 0. All random objects are defined on a probability space P
with expectation operator E·.
The exogenous arrival process Et t ≥ 0 has rate > 0; it is a delayed renewal process starting from zero,
with ith jump time Ui . For t ≥ 0, Et is the number of jobs that arrive to the buffer during 0 t. For i ≥ 1, Ui
is the arrival time of job i and = 1/EU2 is the arrival rate. Jobs already in the buffer at time 0 are called
initial jobs.
For i ≥ 1, the service time Bi is a strictly positive random variable representing the amount of processing
time that job i requires from the server. The random variable Di is strictly positive and determines the deadline
of job i: It represents the maximum amount of time that job i will stay in the buffer. Because job i arrives
at time Ui , its deadline is at time Ui + Di . It will abandon the system at this time if it has not yet completed
service. The random variable Di is called the initial lead time of job i.
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377
The model allows either the service time or the initial lead time (but not both) to be equal to infinity. In
this way, the ordinary PS queue (Di ≡ ) and the infinite server queue (Bi ≡ ) are included as special cases.
2+ = 0 × 0 . Here, + = 0 is the
Therefore, the random vector Bi Di takes values in the space + , x · = for x > 0, and
disjoint union 0 ∪ , with the arithmetic extensions x + = for all x ∈ + = 0 as used in this paper is distinct from the one-point compactification
0 · = 0. (Note that the space 2+ is denoted by . Throughout the paper, it is assumed that
of 0 .) The collection of Borel subsets of all sequences of service times and initial lead times Bi Di are independent and identically distributed (i.i.d.)
2+ -valued random variables, and that their common joint distribution on 2+ satisfies
+ = + × 0 = = 0
0 × (2.1)
2+ with distribution
Note that the random variables Bi and Di may be dependent. A generic random element of will be denoted B D.
Initial condition. The initial condition specifies Z0, the number of initial jobs present in the buffer at time
zero, as well as the service times and initial lead times of these initial jobs. Assume that Z0 is a nonnegative,
integer-valued random variable. The service times and initial lead times for initial jobs are the first Z0 elements
of a sequence Bj0 Dj0 of random variables taking values in 0 × 0 \ almost surely. Assume that
the expected number of initial jobs is finite: EZ0 < .
Time evolution of the queue. For each t ≥ 0, let Zt denote the number of jobs in the buffer (or queue
length) at time t, and let St denote the cumulative service per job provided by the server up to time t. Because
of the processor sharing policy, St is given by
t 1
St =
ds
(2.2)
0 Zs
where the integrand is defined to be zero when the queue length equals zero. If a job arrived at time s ≥ 0 and
is still present in the queue at time t ≥ s, then by time t it has received the cumulative amount of processing
time Ss t = St − Ss.
Therefore, the residual service time at time t of job i ≤ Et and initial job j ≤ Z0 are given by
Bi t = Bi − St − SUi +
Bj0 t = Bj0 − St+ and
(2.3)
Define the lead time at time t of job i ≤ Et and initial job j ≤ Z0 by
Di t = Ui + Di − t+
Dj0 t = Dj0 − t+ and
(2.4)
A job’s residual service time is the remaining amount of processing time required to fulfill its service requirement; its lead time is the remaining time until its deadline. Job i will depart the system either when its service
requirement is fulfilled or when its deadline is reached; it will leave the system at time
inft ≥ Ui " minBi t Di t = 0
The state descriptor is a measure-valued process that keeps track of the residual service times and lead times
2+
of all jobs in the buffer. For job i, this information is represented as a unit of mass at the point Bi t Di t ∈ +
2+
at all times t ≥ Ui such that job i is still in the system. Let #x y denote the Dirac point measure at x y ∈ +
if minx y > 0, otherwise #x y is the zero measure. Then, the state of the system at time t ≥ 0 is represented
by the random point measure
Z0
Et
+
+
t =
#B0 t D0 t +
#Bi t Di t (2.5)
j=1
j
j
i=1
Note that the queue length at time t is given by the total mass of the measure t,
Zt = 1 t
(2.6)
2+ and a &-integrable function f " 2+ → .
where f & = 2 fd& for a Borel measure & on +
2+ moving
In this way, the dynamics of the system are represented as a distribution of point masses on toward the axes. At time t ≥ 0, points move left at rate 1/Zt and down at rate 1. (A point with one coordinate
Gromoll, Robert, and Zwart: Fluid Limits for Processor-Sharing Queues with Impatience
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Mathematics of Operations Research 33(2), pp. 375–402, © 2008 INFORMS
Lead times
Di (t)
1/Z(t)
1
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Service
completion
Bi (t)
Abandonment
Residual service times
Figure 1. Dynamics of the measure-valued process ·.
equal to infinity will remain that way while the other coordinate moves.) Point masses vanish when hitting one
of the axes: a point mass reaching the vertical axis corresponds to a job completing service, while a point mass
hitting the horizontal axis represents a job abandoning the queue (see Figure 1).
2+ , endowed with the topology of weak
Let M denote the space of finite nonnegative Borel measures on w
2+ → . Let
convergence: n → in M if and only if f n → f for all continuous bounded functions f " D0 M denote the space of càdlàg paths in M, endowed with the Skorohod J1 -topology. Then, for t ≥ 0,
t is a random element of M and · is a random element of D0 M.
It is clear that, given stochastic primitives E·, Bi Di , and the initial condition 0, the Equations (2.2)–
(2.6) uniquely determine the processes S·, Z·, ·, and the residual service times and lead times. It is also
easily seen that the state descriptor · satisfies the following equation: For each Borel set A ∈ and all t ≥ 0,
tA = 0A + St t +
Et
i=1
1+
A Bi t Di t
(2.7)
+
where A + w = a + w" a ∈ A and 1+
A w = 1A #w . Note that the quantity 0A + St t corresponds
2+ and s t ∈ 2+ , then for A ∈ ,
to a shift of the initial points by the vector St t: if x y ∈ #x y A + s t = #x−s y−t A
Equation (2.7) plays a crucial role in determining fluid limits for the model.
r · associated
2.2. A fluid scaling. We now introduce a sequence of renormalized stochastic processes r
· will give the fluid limits of
to the solution of the evolution Equation (2.7). The limits in distribution of this model.
Let ⊂ 0 be a sequence increasing to infinity. Suppose that for each r ∈ , there is a stochastic model
as defined in §2.1. That is, for each r ∈ , there are stochastic primitives E r · and Bir Dir , with associated
data r and r and an initial condition r 0. As before, these determine stochastic processes Z r ·, S r ·,
r ·, and residual service times and lead times Bir · Dir · and Bi0r · Di0r ·.
A fluid scaling is applied to each model in the sequence. To obtain nontrivial scaling limits, initial lead times
Dir will be assumed to be of order r. Consequently, they must be scaled by r −1 to keep track of them. For
each r ∈ , let ˘ r ∈ M be the probability measure defined by
˘ r F × G = r F × rG
+ , with the notation rG = r · g" g ∈ G. Note that if Bir Dir = Bi rDi for
for all Borel subsets F G of some sequence Bi Di , then ˘ r is simply the distribution of B1 D1 .
r t ∈ M
For each r ∈ , the fluid scaled state descriptor is defined, for t ≥ 0, as the random measure such that
r tF × G = 1 r rtF × rG
r
+ . Note that this definition scales lead times by a factor r −1 as well. Fluid scaled
for all Borel subsets F , G of versions of the remaining processes are defined as follows: for all r ∈ , t ≥ s ≥ 0 and i = 1 / / / E r rt, let
1
Er t = E r rt
r
1
Z r t = Z r rt
r
Gromoll, Robert, and Zwart: Fluid Limits for Processor-Sharing Queues with Impatience
Mathematics of Operations Research 33(2), pp. 375–402, © 2008 INFORMS
Sr t = S r rt
Sr s t = Sr t − Sr s
1
ir t = Dir rt
D
r
Bir t = Bir rt
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379
2.3. Fluid model. We next introduce a deterministic fluid model satisfying dynamic equations analogous
to (2.7). It will be shown that these equations can be obtained as limits of (2.7) under the above scaling.
Let > 0, let ∈ M be a probability measure satisfying (2.1), and let 0 ∈ M be such that 0 × = 0
+ and 0 + × · are free of atoms in 0 . The fluid model is defined
and such that the projections 0 · × r 0 of the stochastic
from the data 0 , which will arise as limiting values of the parameters r ˘ r + with distribution is denoted B D. Let = EB
model (see Assumptions below). A random vector in denote the traffic intensity of the fluid model. It is assumed throughout that > 1, that is, the server is nominally
overloaded.
Definition 2.1. A continuous function ·" 0 → M is called a measure-valued fluid model solution
for the data 0 if
(i) inf t>a zt > 0 for all a > 0,
(ii) for all A ∈ and t ≥ 0,
t
(2.8)
tA = 0 A + S0 t t + A + Ss t t − s ds
0
where for all v ≥ u ≥ 0,
Su v =
and z· is the total mass function
zt = 1 t =
v
u
1
ds
zs
2+
t dx dy
The function z· is called simply a fluid model solution for 0 .
Note that S0 t may be equal to if z0 = 0, and thus 0 = 0. Both right-hand terms in (2.8) are still
well-defined in this case and the first term equals zero.
Define a class of corner sets
+ = x × y " x y ∈ The sets C ∈ are useful for describing the evolution of fluid model solutions. Because each C of the form
x × y is characterized by the coordinates x y of its corner, it will be convenient to use the notation
def.
&x y = &x × y for any & ∈ M. Then, (2.8) can be rewritten for this class of subsets as follows.
Let z0 = 1 0 . Let B0 D0 be a random vector with distribution 0 /z0 if z0 > 0, and let B0 D0 be the zero
+ , and t ≥ 0,
vector if z0 = 0. Recall that B D is a random vector with distribution . Then, for each x y ∈ a measure-valued fluid model solution satisfies
t
(2.9)
tx y = z0 PB 0 ≥ x + S0 t3 D0 ≥ y + t + PB ≥ x + Ss t3 D ≥ y + t − s ds
0
Because zt = t0 0 for each t ≥ 0, a fluid model solution z· satisfies
t
zt = z0 PB 0 ≥ S0 t3 D0 ≥ t + PB ≥ Ss t3 D ≥ t − s ds
0
(2.10)
Conversely, if ·" 0 → M is a continuous function satisfying (i) of Definition 2.1 and (2.9) for all
+ and t ≥ 0, then · is a measure-valued fluid model solution as the following argument shows.
x y ∈ Let be the set of A ∈ for which (2.8) holds for all t ≥ 0. Clearly, ⊂ because (2.9) holds for all
2+ ∈ because 2+ ∈ ; if An ⊂ satisfies An ↑ A,
+ and t ≥ 0. Observe that is a -system: x y ∈ then A ∈ ; if A1 ⊂ A2 are elements of , then A2 \A1 ∈ . Observe also that is a 4-system: If C1 C2 ∈ ,
then C1 ∩ C2 ∈ . Because ⊂ and the 5-algebra generated by is equal to , it follows that = by
the Dynkin 4-theorem (see, for example, Billingsley [2]). Thus, (2.8) holds for all A ∈ and t ≥ 0, and ·
is a measure-valued fluid model solution.
Likewise, the previous argument shows that if z·" 0 → + is a continuous function satisfying (i) of
Definition 2.1 and (2.10) for all t ≥ 0, then · defined by (2.9) is a measure-valued fluid model solution and
z· is its total mass function.
The first result establishes uniqueness of fluid model solutions under a Lipschitz assumption on the initial
condition 0 . Note that uniqueness of fluid model solutions is equivalent to uniqueness of measure-valued fluid
model solutions.
Gromoll, Robert, and Zwart: Fluid Limits for Processor-Sharing Queues with Impatience
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Mathematics of Operations Research 33(2), pp. 375–402, © 2008 INFORMS
Theorem 2.2. Suppose that there exists a finite constant L such that
0 F × y y ≤ Ly − y
(2.11)
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+ and all y > y ≥ 0. Then, a (measure-valued) fluid model solution for data 0 for all Borel sets F ⊂ is unique.
See §3 for the proof. It will be shown that the fluid model defined above describes the limit in distribution
r ·" r ∈ . The measure-valued fluid model solution · corresponds to the
of the rescaled processes measure-valued state descriptor ·, and the fluid model solution z· is the limit of the queue length process
r · is given below. This result also establishes the
Z·. The main result concerning the convergence of existence of (measure-valued) fluid model solutions. We first state the necessary asymptotic assumptions.
Assumptions .
As r → ,
Er t → t
(2.12)
in distribution for the topology of uniform convergence on compact sets. In particular, r → . Furthermore,
w
˘ r →
(2.13)
0 → 0 in distribution.
(2.14)
r
w
r ·" r ∈ is tight and each weak limit point is
Theorem 2.3. If Assumptions hold, then the sequence almost surely a measure-valued fluid model solution · for the data 0 . If, in addition, Condition (2.11)
r · converges in distribution, as r → , to the unique measure-valued fluid model solution ·.
holds, then The proof appears in §§5 and 6.
2.4. Properties of the fluid model. Despite the quite abstract setting of this paper (measure-valued processes), some concrete and explicit results concerning the fluid model can be obtained. Let 0 satisfy the
assumptions of §2.3.
The following result describes the time equilibrium of the fluid model, that is, its behavior as time tends to
infinity.
Theorem 2.4. Suppose that EB1D= < 1 and EminB D < . Then, as t → , any fluid model
solution zt converges to the unique positive solution z of the fixed point equation
z = Eminz B D
Moreover, any measure-valued fluid model solution t converges in M to the unique measure defined by
t
P B ≥ x + 3 D ≥ y + t dt = Eminz B − x+ D − y+ (2.15)
x y = z
0
for x y ≥ 0.
A heuristic interpretation of the important fixed point equation satisfied by z is as follows: Let Z r denote
the steady state number of jobs in the system. Furthermore, let V r B be the sojourn time of a job if the job
never reneges. Then, the actual sojourn time is given by minV r B Dr , and from Little’s law we get
EZ r = EminV r B Dr Divide both sides of this relation by r and let r → . Because we observe the system in steady state at time 0,
the number of jobs hardly changes and, by a snapshot principle, we conclude that V r = Z r B + or. Furthermore,
we have Dr = Dr. Noting that Z r /r → z then gives the desired equation. In §4, this equation is analyzed to
investigate the qualitative behavior of the model.
Theorems 2.2 and 2.4 are proved in §3, and Theorem 2.3 is proved in §§5 and 6. Note that , which describes
the asymptotic behavior of the measure-valued fluid model, has a simple expression in terms of the solution z
of the fixed point equation.
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Mathematics of Operations Research 33(2), pp. 375–402, © 2008 INFORMS
381
3. Properties of the fluid model. In this section, some basic properties of fluid model solutions are derived.
In what follows, let z· be an arbitrary fluid model solution for data 0 . Note that existence of z· is
guaranteed by Theorem 2.3, which is proved in §§5 and 6. Recall that z· satisfies (2.10):
zt = z0 PB 0 ≥ S0 t3 D0 ≥ t + INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
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If z0 > 0, define
t
0
PB ≥ Ss t3 D ≥ t − s ds
S̃t = infs" S0 s ≥ t
(3.1)
Because zt ≤ z0 + t, S0 t → as t → , implying that S̃t is well-defined for all t. In addition, z0 > 0
implies that S̃t < for all t by property (i) of Definition 2.1 and continuity of z·.
t
Define z̃t = zS̃t. By a change of variables, S̃t = 0 z̃u du and z̃· satisfies the equation
z̃t = z0 PB 0 ≥ t3 D0 ≥ S̃t + 3.1. A maximal solution.
section.
Proposition 3.1.
t
0
z̃uPB ≥ t − u3 D ≥ S̃t − S̃u du
(3.2)
An important monotonicity property of fluid model solutions is proved in this
If EB1D= < 1 and EminB D < , then any fluid model solution is bounded.
Proof. Note, first, that because EminB D < , also EminaB D < for every a ∈ 0 . Define
zt = sup0≤u≤t zu. Note that zt ≤ z0 + t < . Fix t and let u ∈ 0 t. Because Ss u ≥ u − s/zt ,
zu ≤ z0 + 0
u
Pzt B ≥ u − s3 D ≥ u − s ds ≤ z0 + Eminzt B D
which is finite because EminB D < . By taking the supremum over u ∈ 0 t and by dividing both sides
by zt one obtains the relation
1 ≤ z0 /zt + EminB D/zt If zt → , then by monotone convergence one gets the inequality
1 ≤ EB1D= which contradicts the assumption EB1D= < 1. We conclude that zt is bounded. Recall from the definition of fluid model solutions that the distributions of B 0 and D0 are assumed to be free
of atoms.
Proposition 3.2 (Maximal Solution). There exists a fluid model solution z∗ · for 0 that is maximal: for any fluid model solution z· for 0 , the relation zt ≤ z∗ t holds for all t ≥ 0.
n
0
Proof. To define z∗ ·, we first
vdefinen a sequence of functions z · inductively as follows. Let z t = z0 +t
n
and, for n ≥ 0, define S u v = u 1/z r dr for v ≥ u ≥ 0 and
zn+1 t = z0 PB 0 ≥ S n 0 t3 D0 ≥ t + t
0
PB ≥ S n s t3 D ≥ t − s ds
(3.3)
We show that zn+1 t ≤ zn t by induction. The inequality z1 t ≤ z0 t is trivial. Suppose that zn t ≤ zn−1 t.
Then, S n u v ≥ S n−1 u v for all v ≥ u ≥ 0 and, using the fact that tail probabilities are nonincreasing,
zn+1 t = z0 PB 0 ≥ S n 0 t3 D0 ≥ t + 0
≤ z0 PB 0 ≥ S n−1 0 t3 D0 ≥ t + t
PB ≥ S n s t3 D ≥ t − s ds
0
t
PB ≥ S n−1 s t3 D ≥ t − s ds
which equals zn t.
Because zn t is nonincreasing in n and nonnegative for all n, there exists a function z∗ t such that z∗ t =
limn→ zn t. If z· is any fluid model solution for 0 , then zt ≤ z∗ t for all t ≥ 0. This is true because
zt ≤ z0 t and, using an inductive argument as above, zt ≤ zn t for all n. Because we know that at least
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Mathematics of Operations Research 33(2), pp. 375–402, © 2008 INFORMS
one fluid model solution exists (Theorem 2.3), it follows that inf t>a z∗ t > 0 for all a > 0. To show that z∗ ·
is continuous, let t h ≥ 0. Then,
z∗ t + h − z∗ t = lim zn t + h − zn t
n→
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By definition of zn ·,
zn t + h − zn t ≤ z0 Pt ≤ D0 < t + h + 0
t
Pt − s ≤ D < t + h − s + h
The right-side tends to zero as h → 0 because D0 has
no atoms and because 0 Pw ≤ D < w + h dw ≤ h.
v
To show that z∗ · satisfies (2.10), let S ∗ u v = u 1/z∗ r dr for all v ≥ u ≥ 0. Then, S n s t → S ∗ s t
for all t ≥ s ≥ 0 by monotone convergence. Because z∗ · is bounded away from zero on 0 , S ∗ s t is
strictly decreasing in s. Thus, because B and D have at most countably many atoms, there are at most countably
many s such that S ∗ s t is an atom of B or t − s is an atom of D. This implies that PB ≥ S n s t3 D ≥
t − s → PB ≥ S ∗ s t3 D ≥ t − s for almost every s ∈ 0 t. Take the limit as n → in (3.3). By the previous
t
discussion, the integral term converges to 0 PB ≥ S ∗ s t3 D ≥ t − s ds by bounded convergence. Because
B 0 and D0 have no atoms, the first right-hand term converges to z0 PB 0 ≥ S ∗ 0 t3 D0 ≥ t.
We conclude that z∗ t is a maximal fluid model solution because it is continuous and satisfies (2.10) and (i)
of Definition 2.1. 3.2. Convergence of fluid model solutions. In this subsection, we show the convergence of fluid model
solutions to a nontrivial constant z as t → .
Proposition 3.3.
If EB1D= < 1, EminB D < , and >1, then the equation
z = Eminz B D
(3.4)
has a unique solution in 0 .
Proof. The function f " a → EminB aD is nondecreasing and concave on 0 . Note that f a =
EminB aD1D< + EB1D= for a > 0. Therefore, f is continuous on 0 , lima→0 f a =
EB1D= < 1, and lima→ f a = EB > 1. Thus, there exists a0 ∈ 0 such that f a0 = 1. Concavity and monotonicity imply that a0 is unique. Otherwise, f would be constant and equal to 1 after a0 , which
contradicts lima→ f a = EB > 1. We conclude that 1/a0 is the unique solution of (3.4). We are now ready to present the main result of this subsection, concerning the asymptotic behavior of any
fluid model solution zt as t goes to infinity.
Theorem 3.4. Let z· be a fluid model solution for 0 and assume that
EB1D= < 1
EminB D < and
> 1
Then, as t → , zt converges to z , the unique positive solution of the fixed point (3.4).
Proof. It suffices to show that z̄ = lim supt→ zt ≤ z and z = lim inf t→ zt ≥ z We start with the
former. Proposition 3.1 implies that z̄ < . For any 9 > 0, there exists a t9 such that t > t9 implies zt ≤ z̄ + 9.
So, for t > t9 , (2.10) yields
zt ≤ z0 PB 0 ≥ S0 t3 D0 ≥ t + +
0
t9
0
t−t9
PB ≥ Ss t3 D ≥ t − s ds
Pminz̄ + 9B D ≥ s ds
Hence, taking the lim sup on both sides and noting that limt→ Ss t = for each fixed s ≥ 0 yields
z̄ ≤ Eminz̄ + 9B D
Letting 9 ↓ 0, we obtain z̄ ≤ z by the dominated convergence theorem. The lower bound follows by an
analogous argument, after first noting that z > 0 because z· is a fluid model solution. Gromoll, Robert, and Zwart: Fluid Limits for Processor-Sharing Queues with Impatience
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3.3. Uniqueness of fluid model solutions under nonzero initial conditions. The uniqueness of fluid model
solutions is difficult to determine in general. If one looks at the time-changed Equation (3.2) and takes = 0,
one gets an ordinary differential equation (ODE). Uniqueness of solutions to such an ODE can usually only
be established by reducing it to some special case or by imposing a Lipschitz condition. If D ≡ , then (3.2)
reduces to a renewal equation for which uniqueness is known to hold.
Unfortunately, this reduction is not possible for more general D. For technical reasons, one has to use a
Lipschitz condition on the distribution function of B 0 D0 . This condition is probably stronger than necessary,
so that the question of uniqueness in general is still open. For related results in the functional analysis literature,
we refer to Chapter 2 of Hale and Verduyn Lunel [15].
Theorem 3.5. Let 0 be such that z0 > 0 and F0 x y = PB 0 ≥ x3 D0 ≥ y is Lipschitz continuous
+,
in y, that is, there is a finite constant L such that for all x y y ∈ F0 x y − F0 x y ≤ Ly − y Then, a fluid model solution for 0 is unique.
Let z· be a fluid model solution for 0 and let · be the corresponding measure-valued fluid
model solution. Let S̃· be defined as in (3.1) and define z̃t = zS̃t. Recall that z̃· satisfies (3.2). Let
˜ = S̃t. It can easily be shown that for all u v ∈ +,
t
˜
tu
v = z0 PB 0 ≥ u + t3 D0 ≥ v + S̃t + t
0
z̃sPB ≥ u + t − s3 D ≥ v + S̃t − S̃s ds
(3.5)
˜
+ , and t ≥ 0, tu
v is completely determined by z̃t and 0 . Thus, uniqueClearly, for any u v ∈ ˜ on A.
ness of z̃t on a time interval A implies uniqueness of t
We next introduce shifted time-changed fluid model solutions. For t0 t ≥ 0, define z̃t0 t = z̃t0 + t and
S̃t0 u v =
v
u
z̃t0 s ds = S̃t0 + v − S̃t0 + u
It is easily checked that for all t0 t ≥ 0,
˜ 0 t S̃t0 0 t + z̃t0 t = t
t
0
z̃t0 sPB ≥ t − s3 D ≥ S̃t0 s t ds
(3.6)
The idea of the proof is as follows: We take a suitable constant a > 0 and prove first that z̃t is unique on
˜ for t ∈ 0 a. Using this and the shifted Equation (3.6),
0 a. As indicated above, uniqueness carries over to t
˜
we then prove uniqueness for z̃t on the interval a 2a and so forth. This iterative procedure works if tu
v
is Lipschitz in v for all t ≥ 0. This is the content of the following lemma.
+ , and t ≥ 0,
Lemma 3.6. Under the assumptions of Theorem 3.5, we have for all x y y ∈ ˜
˜
tx
y − tx
y ≤ z0 L + y − y Proof. We may assume that y ≤ y . From (3.5) we obtain
˜
˜
tx
y − tx
y ≤ z0 Ly − y + t
0
z̃sPS̃t − S̃s + y ≤ D < S̃t − S̃s + y ds
Noting that z̃sds = dS̃s, we can rewrite the right side as
z0 Ly − y + Because for any # > 0,
0
S̃t
0
Pr + y ≤ D < r + y dr
Pw ≤ D < w + # dw ≤ #
we see that (3.7) can be bounded above by z0 L + y − y .
(3.7)
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Proof of Theorem 3.5. Recall that we have fixed a fluid model solution z·. By the one-to-one correspondence between solutions of (2.10) and (3.2), it suffices to show that z̃· is unique. Let a = 1/2z0 L + 4.
We first show that z̃· is unique on 0 a. For this, suppose that there exists a continuous function h· that
t
satisfies (3.2) for all t ∈ 0 a, with H t = 0 hs ds in place of S̃·. Set 9 = sup0≤t≤a z̃t − ht. Then, for
all 0 ≤ s < t ≤ a,
H t − H s − S̃t − S̃s ≤ 9a
t
Recall that S̃t = 0 z̃s ds. Using (3.2) for both z· and h· together with the Lipschitz assumption, we
obtain for t ∈ 0 a,
z̃t − ht ≤ z0 PB 0 ≥ t3 D0 ≥ S̃t − PB 0 ≥ t3 D0 ≥ H t
t
+ z̃sPB ≥ t − s3 D ≥ S̃t − S̃s − hsPB ≥ t − s3 D ≥ H t − H s ds
0
The first right-hand term is bounded by z0 L9a. The second right-hand term is bounded by
t
t
z̃s − hs ds + z̃sPB ≥ t − s3 D ≥ S̃t − S̃s − PB ≥ t − s3 D ≥ H t − H s ds
0
0
Call the previous two terms IIa and IIb. We have IIa ≤ 9a. To bound IIb, we use the inequality
PB ≥ t −s3D ≥ S̃t− S̃s−PB ≥ t −s3D ≥ H t−H s ≤ PS̃t− S̃s−9a ≤ D < S̃t− S̃s+9a
to obtain (after a change of variables r = S̃t − S̃s)
IIb ≤ S̃t
0
Pr − 9a ≤ D < r + 9a dr ≤ 29a
Putting everything together, we see that for t ∈ 0 a,
z̃t − ht ≤ z0 L9a + 39a ≤ 9/2
which implies that 9 = 0. Hence, z̃· and h· coincide on 0 a, and so z̃· is unique on 0 a.
˜
Suppose now that z̃· is unique on 0 ka for some k ≥ 1. This uniquely determines ka.
By (3.6), z̃·
satisfies
t
˜
(3.8)
z̃ka t = kat
S̃ka 0 t + z̃ka sPB ≥ t − s3 D ≥ S̃ka + t − S̃ka + s ds t ≥ 0
0
We now show that this shifted equation has a unique solution
v on 0 a, implying that z̃· is unique on
0 k + 1a. Suppose h· also satisfies (3.8), with H u v = u hs ds in place of S̃ka u v for all v ≥ u ≥ 0.
Set 9 = supt∈0a z̃ka t − ht. As before, we have
S̃ka t − S̃ka s − H t − H s ≤ t − s9 ≤ a9
Using this, we get as before (this time using Lemma 3.6 for the first term on the right)
z̃ka t − ht ≤ z0 L + a9 + 3a9 ≤ 9/2
Thus, 9 = 0 and z̃· is unique on 0 k + 1a. Iterating this argument completes the proof for all t ≥ 0.
3.4. Uniqueness starting from zero. The result in this subsection can be seen as an extension of a result
in Puha et al. [24], where the case of a PS queue without impatience (D ≡ ) was considered.
Theorem 3.7. Suppose that for each 9 > 0, there is a nonincreasing (in each coordinate) function
2+ → , with F9 0 0 > 0 and 0 ≤ F9 x y ≤ 9, such that the equation
F9 " t
z9 t = F9 S9 0 t t + PB ≥ S9 t − s t3 D ≥ s ds
t
0
where S9 s t = s 1/z9 u du has a unique solution z9 · satisfying inf t>a z9 t > 0 for all a > 0. Then, for
each t ≥ 0, z9 t → z∗0 t as 9 ↓ 0, where z∗0 · is the maximal solution starting from zero.
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385
Proof. As in the proof of Proposition 3.2, z9 · can be written as the pointwise limit limn→ zn9 ·, with
zn9 · recursively defined by z09 t = F9 0 0 + t and
t
n
t
=
F
S
0
t
t
+
PB ≥ S9n t − s t3 D ≥ s ds
zn+1
9
9
9
0
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From this construction, it is easily shown by induction on n that zn9 t is nonincreasing in n and that zn9 t ≥ zn0 t.
Induction on n also yields
lim sup z9 t ≤ lim sup zn9 t = zn0 t
9↓0
Because this holds for any n, and
9↓0
zn0 t → z∗0 t,
we can let n → to obtain
lim sup z9 t ≤ z∗0 t
9↓0
Similarly, for every n ≥ 0,
z∗0 t = lim zn0 t ≤ lim sup zn9 t = z9 t
n→
n→
z∗0 t
We conclude that z9 t ≥
for every 9 > 0, which implies the lower limit and the convergence z9 t →
z∗0 t. Uniqueness of fluid model solutions starting from zero is now a simple corollary.
Corollary 3.8. A fluid model solution starting from zero is unique.
Proof. Let z· be a fluid model solution starting from zero. Define z9 t = zt + 9. Then, z9 · satisfies
the equation
t
(3.9)
z9 t = F9 S9 0 t t + PB ≥ S9 t − s t3 D ≥ s ds
where
0
1
F9 x y = P B≥x+
du3 D ≥ s + y ds
0
9−s zu
Observe that for all 9 > 0, F9 satisfies the assumptions of Theorem 3.7. Moreover, F9 is globally Lipschitz
in the second coordinate (with Lipschitz constant 1). Let B90 D90 be distributed as F9 · ·/F9 0 0. Then, by
Theorem 3.5, with B90 D90 in place of B 0 D0 , (3.9) has a unique solution so that z9 · is uniquely determined
by zt 0 ≤ t ≤ 9. Because F9 x y ≤ 9, we see from the previous theorem that z9 t → z∗0 t as 9 → 0. Also,
z9 t = zt + 9 → zt, because zt is continuous. We conclude that zt = z∗0 t, which implies uniqueness.
9
9
3.5. Analysis of the measure-valued fluid model. Some properties of measure-valued fluid model solutions, which are analogues of properties of fluid model solutions, are gathered in the next theorem.
Theorem 3.9. Let · be a measure-valued fluid model solution for 0 with total mass function z·.
w
(i) Suppose that > 1, EminB D < , and EB1D= < 1. Then, t →
z as t → , where for
each c > 0, the measure c is defined by
+
c x × y = PB ≥ x + sc −1 D ≥ y + s ds x y ∈ 0
and z is the unique positive solution of the fixed point Equation (3.4).
(ii) If Condition (2.11) of Theorem 2.2 holds, then · is unique.
Proof. By Theorem 3.4, zt → z as t → . Continuity of z· and finiteness of z imply that there exists
an M < such that zt ≤ M for all t ≥ 0. This implies that Ss t ≥ t − sM −1 for all s t ≥ 0. We first show
that t" t ≥ 0 is relatively compact in M. For n ∈ , let
An = 0 n × 0 n ∪ × 0 n ∪ 0 n × ∪ 2+ and Acn + x y ⊂ Acn + x y for all x ≥ x and y ≥ y , where Acn denotes the complement.
Clearly, An ↑ By (2.8), for all t ≥ 0 and n ∈ ,
t
tAcn = 0 Acn + S0 t t + Acn + Ss t t − s ds
≤ 0 Acn + 0
0
Acn
= 0 Acn + M Acn + sM −1 s ds
386
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2+ = supt≥0 zt ≤ M, the set t" t ≥ 0 is relatively
Thus, limn→ supt≥0 tAcn = 0. Because supt≥0 t
compact in M (see Kallenberg [17, Theorem A 7.5]). Let ∗ ∈ M be a weak limit point (along a subsequence)
of t as t → , and let C ∈ be a continuity set of ∗ . Then, tC → ∗ C. Because zt → z ,
St − t0 t → t0 /z for each t0 > 0. For all 0 ≤ t0 < t,
t0
tC = 0 C + S0 t t + C + St − s t s ds
+
0
t
t0
C + St − s t s ds
Denote the three right-hand terms by I II III. Because zt → z > 0, S0 t t → , and so the first
+
term converges to zero. Because C ⊂ 2,
t
III ≤ PB ≥ sM −1 3 D ≥ s ds
t0
From this bound, it follows that III → 0 as t0 → . Because the marginals of have at most countably many
atoms, C + s/z s is a continuity set for for almost every s. Because St − s t → s/z on 0 t0 as
t → , the bounded convergence theorem implies that
t0
II → C + s/z s ds as t → 0
So we can take t → and then t0 → to conclude that tC → z C as t → . This implies that
∗ C = z C for all C ∈ that are ∗ -continuity sets, which excludes at most countably many C ∈ (because the marginals of ∗ have at most countably many atoms). Thus, ∗ C = z C for all C ∈ , and so
w
∗ = z and t →
z . This proves (i).
To prove (ii), note that z· is unique by Theorem 3.5 and Corollary 3.8. As discussed in §2.3, uniqueness of
· follows. 4. Applications. In this section, we analyze a number of quantitative properties of the fluid model Equation (2.10). In particular, we investigate the fixed point equation
z = Eminz B D
(4.1)
We treat a number of examples which allow for explicit computations, and also obtain a number of stochastic
ordering results. In addition, we investigate the time-dependent behavior of zt for exponentially distributed
lead times.
Apart from the mean queue length z, we are also interested in the long-term fraction of jobs that leave the
system after successful service completion. Denote this fraction by Ps . It is clear that Ps = PD > z B. Consider
two systems indexed by 1 and 2 such that B2 D2 ≡ B1 aD1 for some a > 0, and such that 1 = 2 . Then,
(with obvious notation) we have
z2 = az1 Ps 2 = Ps 1 Consequently, the fraction Ps is invariant under any rescaling of D.
We now proceed by analyzing a number of special cases. In §4.1, we assume a strong form of dependence.
Section 4.3 assumes that B and D are independent. We give a remarkably simple expression for zt in the case
that D has an exponential distribution. Finally, §4.3 considers an example which can be used as a flow level
model for the integration of elastic and streaming traffic.
4.1. Completely dependent lead times. Consider first the case D = ?B, where ? > 0 is a random variable
(independent of B) reflecting the average service rate expected by a job. In this case, the performance measures
can be determined from the equations (recall that = EB > 1)
z = Emin? z Ps = P? > z Some specific examples:
—? single valued. If we assume that ? = @, then z = min@ z , which implies that z = @ because
> 1. From this, it follows that all jobs leave the system impatiently: Ps = P@ > @ = 0. Observe that when
a job leaves the system, a fraction 1/ of its service time has been processed.
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387
—? two valued. From the previous example, it is clear that Ps > 0 only if some jobs are more patient than
others. In this example, we assume that ? equals @1 with probability p and @2 with probability 1 − p. Take
@2 > @1 . Equation (4.1) now simplifies to
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z = p minz @1 + 1 − p minz @2 From this equation and the properties @2 > @1 > 1, it follows that z > @1 . Furthermore, z < @2 holds if and
only if the equation
z = p@1 + 1 − pz
has a nonnegative solution, which is the case if and only if 1 − p < 1 (i.e., when the offered load of the more
patient jobs does not saturate the system). In this case, we have
z =
p@1
< @2 1 − 1 − p
If the last inequality is not valid or if 1 − p ≥ 1, we must have z ≥ @2 which implies
z = p@1 + 1 − p@2 From the above, we can conclude that Ps = 0 if and only if 1 − 1 − p@2 ≤ p@1 . If the reverse inequality
holds, then all of the more patient jobs are being served successfully, i.e., Ps = 1 − p.
—? exponentially distributed. Assume that the mean of ? equals 1. In this case, z can be determined from
the equation z = 1 − z + 1e−z and Ps = e−z .
Because Ps does not depend on the mean of ? and because constant ? yields the worst case Ps = 0, it seems
natural to conjecture that the fraction of successful completions is positively correlated to the variability of ?.
cvx
Thus, it seems worthwhile to look for ordering relations for Ps if ?1 ≥ ?2 . If E?1 = E?2 , this is equivalent
cvx
to Eminx ?1 ≤ Eminx ?2 for all x ≥ 0. So ?1 ≥ ?2 implies that z2 ≥ z1 , that is, less variability
in reneging behavior implies a lower service rate. To prove that P?1 > z1 ≥ P?2 > z2 as well seems
difficult without imposing further assumptions.
4.2. Independent lead times.
In this case, we can write (4.1) as
0
PB > uPD > z u du = 1
which, in case EB < , is equivalent to PD > z B ∗ = 1/, with B ∗ a random variable with density
PB > x/EB which is also independent of D.
Recall that Ps = PD > z B. Consequently, if B is exponentially distributed, we have the insensitivity result
(with respect to the distribution of D) Ps = 1/. The inequality Ps ≤ 1/ holds if B ∗ is stochastically dominated
by B, and Ps ≥ 1/ if the opposite is true. Because B ∗ being stochastically dominated by B is related to low
variability of B, we see again that more variability (this time in the service times) leads to a better system
performance (i.e., higher Ps ).
Exponential reneging. If we assume that D has an exponential distribution with parameter B (and B a general
distribution), we see that z is the solution of
C∗ z B = 1
(4.2)
∗
with C∗ s = Ee−sB . In addition, we have the following remarkable expression for the complete fluid limit
z·, if z0 = 0:
Proposition 4.1. Suppose PD > t = e−Bt , that B is independent of D, and that z0 = 0. Then, the unique
fluid model solution is given by
zt = z 1 − e−Bt (4.3)
with z the solution of (4.2).
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Proof. Recall that (2.10) has a unique solution. We show that (4.3) is indeed the solution of (2.10) by
verification. We thus compute the right-hand side of (2.10), writing zu = z 1 − e−Bu .
Observe that
t 1
1
z
du = logeBt − 1 − logeBs − 1
B
s zu
Consequently,
t
t
t
PD ≥ t − sP B ≥ 1/zu du ds = e−Bt
Pz BB ≥ logeBt − 1 − logeBs − 1 deBs
B
0
s
0
= e−Bt
e−v Pz BB ≥ logeBt − 1 + v dv
B
− logeBt −1
Pz BB ≥ ve−v dv
= e−Bt eBt − 1
B
0
= z 1 − e−Bt C∗ z B = z 1 − e−Bt which shows that z 1 − e−Bt satisfies (2.10). 4.3. TCP-friendly traffic.
means, such that
Assume that there exist independent random variables B1 and D1 , with finite

B1 with probability p
B D =

D1 with probability 1 − p
When we view PS as a way of modeling TCP, this example models the integration of elastic (TCP) traffic
and TCP-friendly user datagram protocol (UDP) traffic; see Key et al. [19] for a related model. The latter type
of traffic is using the system for a certain amount of time, regardless of the level of congestion.
The fixed point Equation (4.1) specializes to
z = pEz B1 + 1 − pED1 Consequently, if the stability condition pEB1 < 1 is satisfied, we see that
z =
1 − pED1 1 − pEB1 5. Tightness. In this section, we prove the first part of Theorem 2.3. That is, we show that the sequence
r · r ∈ is tight in D0 M. The main results in this section implying this property
of processes are the compact containment result in Lemma 5.2 and an oscillation inequality in Lemma 5.6. To prove these
results, a number of further lemmas are developed. Section 5.1 derives a Glivenko-Cantelli theorem for the
r ·. The
stochastic primitives. Section 5.2 introduces a fluid scaled version of the dynamic equation for compact containment property is derived in §5.3. Section 5.4 serves as a preparation for the oscillation bound.
r t charges arbitrarily small mass to thin L-shaped sets. The oscillation bound
In particular, it is shown that is then shown in §5.5.
Throughout this section, it is assumed that Assumptions hold.
5.1. A Glivenko-Cantelli theorem. An important preliminary result is the following functional
Glivenko-Cantelli theorem for the stochastic primitives. It will be used in §§5.3–5.5. It is convenient to consider
the primitives together as a single, measure-valued arrival process. For r ∈ and t ≥ s ≥ 0, define the fluid
scaled measure-valued arrival process by
r
r E t
r t = 1
# r r −1 r i=1 Bi Di r and define the fluid scaled increment
r s t = r t − r s
(5.1)
r s t is a random element
r · is a random element of D0 M and for each t ≥ s ≥ 0, Note that of M.
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To state and prove the result, we first introduce some notions from empirical process theory. Our primary
2+ shatters an n-point subset
reference is van der Vaart and Wellner [28]. A collection of subsets of 2+ if the collection C ∩ x1 xn " C ∈ has cardinality 2n . In this case, say that picks
x1 xn ⊂ out all subsets of x1 xn . The Vapnik-Červonenkis index VC-index of is
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V = minn" shatters no n-point subset
where the minimum of the empty set equals infinity. The collection is a Vapnik-Červonenkis class VC-class
if it has finite VC-index.
2+ and,
VC-classes satisfy a useful entropy bound. Let denote the set of Borel probability measures on 2
for Q ∈ , let f Q = f Q denote the L1 Q-norm of a Borel measurable function f " + → . For 9 > 0,
the L1 Q 9-ball around f is the set of Borel functions g" f − gQ < 9. For a family of functions , the
9 L1 Q-covering number N 9 L1 Q is the smallest number of L1 Q 9-balls needed to cover . If is a VC-class, then for all 9 > 0, the family = 1C " C ∈ satisfies
sup log N 9 L1 Q < 3
(5.2)
Q∈
see Theorem 2.6.4 in van der Vaart and Wellner [28].
Recall the collection of corner sets defined in §2.3:
+ = x × y " x y ∈ 2+ , it is impossible for to pick out all three 2-point subsets of
Note that for any 3-point subset x1 x2 x3 ⊂ x1 x2 x3 . Because shatters no 3-point subset, it has VC-index bounded above by 3. Thus, is a VC-class
and = 1C " C ∈ satisfies (5.2).
2+ → + be the map 4x y = maxx y. Because
Define an envelope function for as follows. Let 4" + -valued random
4 is continuous, (2.13) and the Skorohod representation theorem imply the existence of variables X r with distribution ˘ r 4 −1 and X with distribution 4 −1 such that X r → X almost surely. Thus,
+ -valued random variable Y such that, almost surely,
there exists an Y = sup X r (5.3)
r∈
+ . Because L2 & contains continuous unbounded functions, there exists a continuous,
Let & be the law of Y on + → + that is increasing on 0 , satisfies G ≥ 1, and such that G 2 & < . This
unbounded function G" implies that
G 42 = EGX2 ≤ EGY 2 < (5.4)
Let F = G 4, and note that 1C ≤ F for all C ∈ . That is, F is an envelope function for . Finally, define
= ∪ F .
Lemma 5.1. Let T > 0. Then, as r → ,
r
P
r s t − r t − sf ˘ r →
0
sup sup f f ∈
0≤s≤t≤T
(5.5)
Proof. Let 9 > 0. By (5.1), it suffices to show that
r t − r tf ˘ r > 9 ≤ 9
lim sup Pr sup sup f r→
f ∈
t∈0 T Note that the above event is measurable for each r because it can be rewritten using the suprema over rational t,
r t and r tf ˘ r and f = 1C with C having rational or infinite corner coordinates x and y. Because f are nondecreasing in t for each fixed f ∈ , it suffices to show that for each fixed t ∈ 0 T ,
r
r
r
r
˘
lim sup P sup f t − tf > 9 ≤ 9
r→
Because
f ∈
r t
f r
r
r
r
r
r
r
r
˘
˘
˘
f t − tf = f E t − t + E t
− f Er t
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(with the convention that division by zero equals zero), it suffices to show the two bounds
9
9
r
r
r
˘
lim sup P sup f E t − t >
≤ 2
2
r→
f ∈
r t
9
9
f lim sup Pr supEr t
≤ − f ˘ r >
r
2
2
E t
r→
f ∈
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r
(5.6)
The first inequality follows from assumption (2.12) and by observing that
sup supf ˘ r ≤ supF ˘ r = sup EGX r ≤ EGY < r∈ f ∈
r∈
r∈
(5.7)
which follows from (5.3) and (5.4). To show (5.6), it suffices to verify three assumptions of Theorem 2.8.1 in
van der Vaart and Wellner [28]. Observe that for each n ∈ and e1 en ∈ n , the function
x1 xn → sup
n
f ∈
i=1
ei f xi 2+ ˘ r n for each r ∈ . Thus, is a ˘ r -measurable class for each
is measurable on the completion of r ∈ ; see Definition 2.3.3 in van der Vaart and Wellner [28]. Moreover, is uniformly bounded above by the
envelope function F , and
lim supF 1F >M ˘ r = 0
M→ r∈
by Markov’s inequality, (5.3), and (5.4). Lastly, satisfies the finite entropy bound (5.2) because
N 9 L1 Q ≤ N 9 L1 Q + 1 and is a VC-class. The previous three observations imply that the
assumptions of Theorem 2.8.1 in van der Vaart and Wellner [28] are satisfied. Consequently, is GlivenkoCantelli, uniformly in r. That is, for every # > 0, there exists an n# such that n ≥ n# implies
m
1 sup Pr sup sup
f Bir Dir r −1 − f ˘ r > # ≤ #
m≥n f ∈
m i=1
r∈
(5.8)
Choose # = min9/2 9/4T . The left side of (5.6) is bounded above by
r t
f 9
r
r
r ˘
lim sup P E t > 2T + lim sup P sup
− f >
4T
Er t
r→
r→
f ∈
r
The first term equals zero by (2.12). For the second term, rewrite
r rt
r t
f 1 E
f Bir Dir r −1 = r
E rt i=1
Er t
and bound each probability in the second term by
m
1
9
r
r −1
r ˘
P E rt < n# + P sup sup
f Bi Di r − f >
4T
m≥n# f ∈
m i=1
r
r
r
(5.9)
By (2.12), the first term in (5.9) converges to zero as r → . By (5.8), the second term is bounded above by
# ≤ 9/2, uniformly in r ∈ . This implies (5.6). Gromoll, Robert, and Zwart: Fluid Limits for Processor-Sharing Queues with Impatience
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5.2. Fluid scaled dynamic equation. Using (2.7), it is easy to see that the fluid scaled state descriptor of
the rth model satisfies the following equation almost surely: for each Borel set A ∈ , and all t h ≥ 0,
r E t+h
r t + hA = r tA + Sr t t + h h + 1
ir t + h
1+ Br t + h D
r i=r Er t+1 A i
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r
(5.10)
Subsequent proofs use estimates obtained from this equation. Two estimates result from bounding the summands in (5.10) by one and optionally bounding the first term on the right side by its total mass; for each A ∈ and t h ≥ 0,
r t t + h
r tA + Sr t t + h h + r t + hA ≤ 2+ r t t + h
r t
2+ + 2+ ≤
(5.11)
Two more estimates follow from (5.10) by simply ignoring any arrivals; for each A ∈ and t h ≥ 0,
r t + hA ≤ r t + h
r tA + Sr t t + h h ≤ 2+ (5.12)
5.3. Compact containment. This section establishes the compact containment property needed to prove
tightness.
Lemma 5.2. Let T > 0 and K > 0. There exists a compact set K ⊂ M such that
r t ∈ K for all t ∈ 0 T ≥ 1 − K
lim inf Pr (5.13)
r→
2+ < , and if there exists a sequence of nested
Proof. A set K ⊂ M is relatively compact if supL∈K L
2+ such that n∈ Kn = 2+ and
compact sets Kn ⊂ lim sup LKnc = 0
n→ L∈K
where Knc denotes the complement of Kn ; see Kallenberg [17, Theorem A 7.5.] Consider the nested sequence
2+ given by
of compact sets in Kn = 0 n × 0 n ∪ 0 n × ∪ × 0 n ∪ × n ∈ w
r 0 is tight. Thus, there is a compact set
r 0 →
0 in distribution, and so the sequence By (2.14), K0 ⊂ M such that
r 0 ∈ K0 ≥ 1 − K lim inf Pr (5.14)
r→
2
2+ , and let an = supL∈K LKnc for each n ∈ . Because K0 is compact, M0 < and
Let M0 = supL∈K0 L
0
2+ such that n∈ Jn = 2+ and limn→ supL∈K LJnc = 0.
there exists a sequence of nested compact sets Jn ⊂ 0
Because Jn ⊂ Kkn for each n ∈ and sufficiently large kn ∈ , it follows that an → 0 as n → .
Recall the definition from §5.1 of the envelope function F = G 4 for the family . By (2.12) and (5.7), the
constant M = supr∈ r T F ˘ r + 1 is finite. Let K be the closure of the set
2+ ≤ M0 + M and LKnc ≤ an + Gn−1 M for all n ∈ L ∈ M" L
Because an + Gn−1 M → 0 as n → , the set K is compact in M.
For each r ∈ , denote the event in (5.14) by r0 and define the event
r T ≤ r T F ˘ r + 1
r1 = F By (5.14) and Lemma 5.1, lim inf r→ Pr r0 ∩ r1 ≥ 1 − K. Fix N ∈ r0 ∩ r1 and t ∈ 0 T , and assume for the
r t ∈ K.
remainder of the proof that all random objects are evaluated at this N. Then, it suffices to show that By (5.11),
r t
r 0
r t
2+ ≤ 2+ + 2+ r t ≤ 1 r T ≤ F r T , the definitions of r0 , r1 , and M imply that
r t
2+ = 1 Because r t
2+ ≤ M0 + M
(5.15)
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Fix n ∈ . By (5.10),
r 0Knc + Sr 0 t t + 1
r tKnc = r
The shape of the set
Knc
i=1
r
r
1+
Knc Bi t Di t
implies that
Knc + S0 t t ⊂ Knc
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r Er t
for i = 1 r Er t. Thus,
and
r
r −1
r
r
1+
Knc Bi t Di t ≤ 1Knc Bi Di r r t
r tKnc ≤ r 0Knc + 1K c n
By definition of G, F , and by Markov’s inequality, 1Knc ≤ Gn−1 F . So,
r t
r 0Knc + Gn−1 F r tKnc ≤ r T , the definitions of r0 , r1 , and M imply that
r t ≤ F Because F r tKnc ≤ an + Gn−1 M
(5.16)
r t ∈ K. Equations (5.15) and (5.16) imply that 5.4. Asymptotic regularity. The second and main step necessary to prove tightness is to bound the prob
r · oscillates. Oscillations may result from sudden arrivals or departures of a large
ability that the process amount of mass. Sudden arrivals are controlled by the regularity of the arrival process. To show that sudden
r · assigns arbitrarily small mass to the boundaries of the sets
departures are unlikely as well, we show that C ∈ . This is phrased in terms of O-enlargements of the boundaries of these sets (forming a collection of
2+ and let
L-shaped sets). For C ∈ and O > 0, let PC denote the boundary of C in 2+ " inf w − z < O
PCO = w ∈ z∈PC
2+ of its boundary, where the infimum over the empty set equals . (Note that PC
be the O-enlargement in O
2+ . Note also that PCO = x − O+ x + O × for a
and, therefore, also PC is empty for the corner set corner set of the form x × with x ∈ 0 .) The following lemma establishes the result for the initial
r 0.
condition Lemma 5.3.
For all 9 K > 0, there exists a O > 0 such that
r 0PCO ≤ 9 ≥ 1 − K
lim inf Pr sup r→
Proof.
O > 0,
C∈
(5.17)
r1 0· = r 0· × r2 0· = r 0
+ and + × ·. For each C ∈ and
Fix 9 K > 0 and let + ∪ + × y y + 2O
PCO ⊂ x x + 2O × for some x y ∈ 2+ = 0 × 0 . Thus, it suffices to show that for i = 1 2, there exists a O > 0 such that
ri 0x x + 2O ≤ 9 ≥ 1 − K (5.18)
lim inf Pr sup r→
2
2
x∈0
We prove the statement for i = 1; the proof is identical for i = 2.
r1 0 converges in distribution to 0 · × +
The projection x y → x is continuous, so (2.14) implies that + is free of atoms in 0 , there exists a O > 0 such that
as r → . Because 0 · × + ≤ 9 sup 0 x x + 4O × 2
x∈0
(5.19)
+ .) Moreover, there exists a constant M such that
(If (5.19) fails, it is easy to construct an atom of 0 · × + ≤ 9 0 M × 2
(5.20)
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Let N = M/O + 1, where x denotes the smallest integer n ≥ x. For n = 0 / / / N − 1, define the set
In = nO n + 4O and define IN = M . Note that for every x ∈ 0 , there is an n ≤ N such that
x x + 2O ⊂ In . To prove (5.18), it therefore suffices to show that
9
K
r
r
≥1− (5.21)
lim inf P max 1 0In ≤
r→
n≤N
2
2
+ , endowed with the weak topology.
+ denote the space of finite nonnegative Borel measures on Let M
w
+ " maxn≤N LIn < 9/2, and suppose that a sequence Lk ⊂ M
+ satisfies Lk →
Let A = L ∈ M
L for
some L ∈ A. Because the sets In are closed, the Portmanteau theorem (adapted to finite measures) implies that
lim sup Lk In ≤ LIn <
k→
9
2
for all n ≤ N + . Thus, a second application of
Hence, Lk ∈ A for sufficiently large k, which implies that A is open in M
the Portmanteau theorem yields
r1 0 ∈ A ≥ P0 · × + ∈ A = 1
lim inf Pr r→
which implies (5.21). r ·.
The regularity result is now shown for the entire state descriptor Lemma 5.4.
Let T > 0 and 9 K > 0. There exists a O > 0 such that
r
r
O
lim inf P sup sup tPC ≤ 9 ≥ 1 − K
r→
C∈ t∈0T (5.22)
Proof. By Lemmas 5.1, 5.2, and 5.3, there exists a compact K ⊂ M and a O0 > 0, such that for all # > 0,
the events
r 0PCO0 ≤ 9 r1 = sup 2
C∈
r s tC − r t − s˘ r C ≤ # r2 = sup sup C∈ 0≤s≤t≤T
r
t ∈ K for all t ∈ 0 T r3 = r0 = r1 ∩ r2 ∩ r3 satisfy
lim inf Pr r0 ≥ 1 − K
(5.23)
r→
Recall the compact sets Kn defined in the proof of Lemma 5.2. Because K is compact, there exists a finite
M ≥ 1 and an integer R < such that
2+ ≤ M
sup L
(5.24)
9
sup LKRc ≤ 2
L∈K
(5.25)
L∈K
Let ∗ = supr∈ r , which is finite by (2.12). Fix
h = 98∗ −1 O = minO0 h2M−1 and
# = 9 min8RMh−1 −1 2−1 For r ∈ , let r∗ denote the event in (5.22). By (5.23), it suffices to show that r0 ⊂ r∗ . Let N ∈ r0 be
arbitrary; for the remainder of the proof, all random objects are evaluated at this N.
r tPCO ≤ 9. Define the random time
Consider any r ∈ , t ∈ 0 T , and C ∈ . We must show that r s = 0
R1 = sups ≤ t" 1 if the supremum exists, and define R1 = 0 otherwise. Let R = maxR1 t − RM. We first show that
r RPCO + Sr R t t − R ≤ 9 2
(5.26)
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If R = 0, this follows from the definition of r1 because O ≤ O0 , because
O
PCO + Sr R t t − R ⊂ PC+
Sr R t t−R
and because is closed under positive translation. Suppose R = R1 > 0. Then, there is a sequence Rn , with
r Rn = 0 for all n. In this case, (5.11) and the definition of r2 imply that, for all n,
Rn ↑ R, such that 1 INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
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r
r
r Rn r RPCO + Sr R t t − R ≤ +
+
2 + Rn R2 ≤ R − Rn + #
Letting Rn ↑ R yields
r RPCO + Sr R t t − R ≤ # ≤ 9 2
r
Suppose that R = t − RM. Because 1 s > 0 for all s ∈ R t, the definition of r3 and (5.24) imply that
t
r s−1 ds ≥ R
1 Sr R t =
t−RM
Thus, by the definition of r3 and (5.25),
r RPCO + Sr R t t − R ≤ r RKRc ≤ 9 2
which proves (5.26).
By (5.10),
1 r Et + r
r
O
r
O
r
ir t
tPC = RPC + S R t t − R +
1 O B t D
r i=r Er R+1 PC i
r
(5.27)
Let I denote the second right-hand term in (5.27). By (5.26), it remains to show that I ≤ 9/2. Let N =
t − Rh−1 and, for each n = 0 / / / N − 1, let tn = R + nh and t n = mintn+1 t. Then, using the inequality
1+
PCO · · ≤ 1PCO · ·,
I≤
N
−1
n=0
t 1 r E
ir t
1 O Br t D
r i=r Er t +1 PC i
r
n
(5.28)
n
Consider n ∈ 0 / / / N − 1 and i such that Uir r −1 ∈ tn t n . Observe that
By definition,
Sr t n t ≤ Sr Uir r −1 t ≤ Sr tn t
(5.29)
ir t = 1PO +Sr U r r −1 t t−U r r −1 Bir Dir r −1 1PCO Bir t D
i
i
C
(5.30)
So, letting
+
Cn− = C + Sr t n t − O t − t n − O ∩ 2
+
Cn+ = C + Sr tn t + O t − tn + O ∩ 2
Cn = Cn− \Cn+ it follows from (5.29) and (5.30) that
ir t ≤ 1C Bir Dir r −1 1PCO Bir t D
n
Conclude from (5.28) and (5.31) that
I≤
N
−1
n=0
For all n < N ,
Cn− Cn+
t N
−1
1 r E
r tn t n Cn− − r tn t n Cn+ 1Cn Bir Dir r −1 =
r i=r Er t +1
n=0
r
n
n
n
∈ and t − tn ≤ h. So, the definition of r2 implies that
I≤
N
−1
n=0
r h˘ r Cn + 2#
(5.31)
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395
By definition of N , and because t − R ≤ RM,
I ≤ ∗ h
N
−1
n=0
˘ r Cn + RMh−1 2#
This implies, by choice of #, that
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I ≤ ∗ h
If n ∈ 0 / / / N − 3, then
N
−1
9
˘ r Cn + 4
n=0
(5.32)
Sr tn+1 tn+2 ≥ hM −1 ≥ 2O
r s ≤ M for all s ∈ R t and because h ≥ O2M by definition of O. Thus, for all n ∈
because 0 < 1 0 / / / N − 3,
Sr t n t − O = Sr tn+1 tn+2 + Sr tn+2 t − O ≥ Sr tn+2 t + O
+
for all n ∈ 0 / / / N −3 and, consequently, Cn ∩Cn+2 = !. Thus, because ˘ r is a probability
Hence, Cn− ⊂ Cn+2
measure,
"N −1/2#
"N −2/2#
and
˘ r C2n ˘ r C2n+1 n=0
n=0
are both bounded above by one. Conclude from (5.32) that
9
I ≤ 2∗ h + 4
which implies, by choice of h, that I ≤ 9/2. 5.5. Oscillation bound. This section establishes the second main ingredient for proving tightness of the
state descriptors. As a metric on M, we use the Prohorov metric (adapted to finite measures). For & B ∈ M,
define
d& B = inf9>0" &A≤BA9 + 9 and BA ≤ &A9 + 9 for all closed A ∈ 2+ " inf z∈A z − w < 9 and that denotes the Borel subsets of 2+ .
Recall that A9 = w ∈ Definition 5.5. For each · ∈ D0 M and each T > # > 0, define the modulus of continuity on
0 T by
wT · # = sup sup dt + h t
t∈0 T −# h∈0 #
Lemma 5.6. For all T > 0 and 9 K ∈ 0 1, there exists # ∈ 0 T such that
r · # ≤ 9 ≥ 1 − K
lim inf Pr wT r→
(5.33)
Proof. Fix T > 0 and 9 K ∈ 0 1, and let ∗ = supr∈ r For each O > 0, define
+ ∪ + × 0 O
LO = 0 O × By Lemmas 5.1 and 5.4, there exists O ∈ 0 1 such that for all # ∈ 0 T , the events
9
r
r
1 = sup tLO ≤
4
t∈0 T r
r
2
∗
2 =
sup t t + #+ ≤ 2 # t∈0 T −#
r0 = r1 ∩ r2 satisfy
lim inf Pr r0 ≥ 1 − K
r→
(5.34)
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Fix # = O92 8 max∗ 1−1 and let r∗ be the event in (5.33). By (5.34), it suffices to show that r0 ⊂ r∗ for
each r. Fix r ∈ and N ∈ r0 ; for the remainder of the proof all random objects are evaluated at this N. Fix
t ∈ 0 T − #, h ∈ 0 # and let A ∈ be closed. It suffices to show the two inequalities,
r tA ≤ r t + hA9 + 9
and
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r t + hA ≤ r tA9 + 9
(5.35)
(5.36)
To show (5.35), use the definition of r1 to write
r tA ≤ r tLO + r tA ∩ LcO 9 r
≤ +
tA ∩ LcO (5.37)
4
r s ≥ 9/4 for all s ∈ t t + h, which
r s < 9/4. Suppose I = !. Then, 1 Let I = s ∈ t t + h " 1 implies that
t+#
r s−1 ds + # ≤ 4# + # < min9 O
1 (5.38)
Sr t t + h h ≤
9
t
Consequently, x y ∈ A ∩ LcO implies x y − Sr t t + h h ∈ A9 , and so
A ∩ LcO ⊂ A9 + Sr t t + h h
Deduce from (5.37) that
(5.39)
r tA ≤ 9 + r tA9 + Sr t t + h h
4
Apply (5.12) to get
r t + hA9 r tA ≤ 9 + (5.40)
4
r s ≥ 9/4 for all
r R ≤ 9/4 by right continuity. Because 1 Suppose I =
$ ! and let R = inf I. Then, 1 s ∈ t R,
R
r s−1 ds + # ≤ 4# + # < O
(5.41)
Sr t R R − t ≤ 1 9
t
By (5.37) and (5.41),
r tLcO ≤ 9 + r t
r tA ≤ 9 + 2+ + Sr t R R − t
4
4
Apply (5.12) to get
r R
r tA ≤ 9 + 2+ ≤ 9 (5.42)
4
2
So, (5.35) follows because either (5.40) or (5.42) holds.
To show (5.36), use (5.11) and the definitions of r2 and # to obtain
r t t + h
r t + hA ≤ r tA + Sr t t + h h + 2+ r tA + Sr t t + h h + 9 ≤
4
r
9
If I = !, then (5.38) implies that A + S t t + h h ⊂ A . So, (5.43) yields
r tA9 + 9 r t + hA ≤ 4
r
If I $= !, then by (5.11), the definition of 2 , and the choice of #,
r R t + h
r R
r t + hA ≤ 2+ + 2+ ≤ 9 + 2∗ # ≤ 9 4
2
In both cases, (5.36) holds. Conclude from (5.35) and (5.36) that
r t r t + h ≤ 9
d
Because t ∈ 0 T − # and h ∈ 0 # were arbitrary,
r · # ≤ 9
wT which implies that N ∈ r∗ .
(5.43)
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397
6. Limiting fluid equations. This section contains the proof of Theorem 2.3. Tightness of the sequence
r · follows immediately from Lemmas 5.2 and 5.6. Because r · is tight, there exists a subsequence
q · ⇒ · as q → . We must show that · is
q ⊂ and a process · in D0 M such that almost surely a measure-valued fluid model solution for the data 0 . This is accomplished in Lemmas
6.1 and 6.2 and Theorem 6.3 below. Finally, if (2.11) holds, then a measure-valued fluid model solution for
r · ⇒ ·
0 is unique by Theorem 2.2. In this case, the law of the limit point · is unique and so as r → .
v
Let Z· = 1 · be the total mass process for ·, and let Su v = u 1/Zs ds for all v ≥ u ≥ 0.
To show that · is almost surely a measure-valued fluid model solution, note first that · is almost surely
continuous by Lemma 5.6. Note also that, by (2.14), 0 = 0 almost surely. It remains to show that properties
(i) and (ii) of Definition 2.1 are satisfied almost surely by ·. The next result establishes (i).
Lemma 6.1. Almost surely, for all a > 0,
inf Zt > 0
t>a
(6.1)
Proof. Suppose first that PB = > 0, and let T > a > 0 be arbitrary. It suffices to show that
inf t∈aT Zt > 0 almost surely. Assume without loss of generality that a is sufficiently small that PB = 3
D ≥ a > 0, and that the distribution of D is continuous at a. Define the corner set Ca = × a and let
ka = aCa /2 > 0. For each q ∈ and t ∈ a T ,
q t
q t × 0 2+ ≥ Z q t = Applying (5.10) and then dropping the first right-hand term yields
q
1 q Et
q
1+
×0 Bi t Di t q i=q Eq t−a+1
q
Z q t ≥
(6.2)
iq t = Diq q −1 − t + Uiq q −1 ≥ Diq q −1 − a. This
Note that for i > q Eq t − a, we have Uiq ≥ qt − a and so D
q
q
q
+
i t ≥ 1C Bi Diq q −1 . Deduce from (6.2) that
implies that, for each such i, 1×0 Bi t D
a
q t − a tCa Z q t ≥ (6.3)
Because Ca is a -continuity set, q a˘ q Ca → aCa by (2.12) and (2.13). So, by (6.3), Lemma 5.1, and
the definition of ka ,
q t − a tCa ≥ ka = 1
lim inf Pq inf Z q t ≥ ka ≥ lim inf Pq inf (6.4)
q→
t∈a T q→
t∈a T q · ⇒ · implies Z q · ⇒ Z·, and that the set
Note that z· ∈ D0 + " inf zt ≥ ka
t∈aT is closed in the Skorohod J1 -topology. Thus, the Portmanteau theorem and (6.4) imply that
q
q
P inf Zt ≥ ka ≥ lim inf P
inf Z t ≥ ka = 1
t∈aT q→
t∈aT It remains to consider the case PB = = 0. Again, we will construct a lower bound on the queue length
process. The main difference with the case above is that we exploit results for overloaded PS queues without
impatience developed in Puha et al. [24]. Because the idea of the lower bound is similar as before, we restrict to
giving an outline. If PB = = 0, there exists m < such that EB1B≤m > 1, and such that the distribution
of B is continuous at m. Fix a > 0, and assume without loss of generality that a is sufficiently small that
EB1B≤m3D>a > 1, and that the distribution of D is continuous at a. Then, by (2.13),
lim q B1q 1B1q ≤m3 D1q q −1 >a = EB1B≤m3 D>a q→
Compare Z q · with the queue length process Ź q · of an ordinary PS queue having arrival rate qa m =
q PB1q ≤ m3 D1q > qa and service times Biq a m , which are distributed as Biq Biq ≤ m3 Diq > qa. This process
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can be constructed from the primitives E q ·, Biq Diq in an obvious way, enabling a sample-path comparison
of the two processes. Assume that Ź q qt − a = 0.
Observe that the number of arrivals in the ordinary PS queue during the time interval qt − a qt is less
than or equal to the number of arrivals during that time interval in the PS queue with impatience. Furthermore,
if a job that arrived in the original system after time qt − a departs before time qt, then this must also be
the case in the ordinary PS queue, because that system had a service rate which was at least as large as in
the original system. These considerations imply that Z q qt ≥ Ź q qt. The ordinary queue is still overloaded
(for sufficiently large q), no customer departs because of impatience, and the modified arrival process is still a
renewal process. Thus, the evolution of the modified system during qt − a qt has the same law as that of
an overloaded GI/GI/1 PS queue starting at 0, in the time interval 0 qa.
Because the service times in our modified system are bounded, the means converge as q → . The assumptions in Puha et al. [24] are therefore satisfied, and it follows that there exists a constant ka > 0 such that
limq→ Ź q qt/q = ka almost surely. Consequently, we have lim inf q→ Z q t ≥ ka almost surely, which implies
the assertion. Before establishing property (ii) of Definition 2.1, the following result is needed.
Lemma 6.2. Almost surely, for all C ∈ and t ≥ 0,
tPC = 0
(6.5)
Proof. Let T > 0. It suffices to show the statement for all t ∈ 0 T . Let Kn ⊂ 0 1 be a sequence such
that n=1 Kn < . By Lemma 5.4, there exists a null sequence of positive reals On such that, for each fixed n,
q tPCOn ≤ 1 ≥ 1 − Kn lim inf Pq sup sup (6.6)
q→
n
t∈0 T C∈
O
For each n ∈ , let Mn = L ∈ M" supC∈ LPCn ≤ 1/n. If a sequence Li ⊂ Mn converges weakly to L, then
O
for each open set PCn the Portmanteau theorem yields
1
O
O
LPCn ≤ lim sup Li PCn ≤ n
i→
Thus, L ∈ Mn and Mn is closed. By definition of the Skorohod J1 -topology, the set DTn = · ∈
D0 M" t ∈ Mn for all t ∈ 0 T is also closed. Apply the Portmanteau theorem and (6.6) to obtain
q · ∈ DTn ≥ 1 − Kn P· ∈ DTn ≥ lim inf Pq q→
By the Borel-Cantelli lemma,
P
k=1 n=k
· ∈ DTn = 1
Thus, there exists a finite random variable N such that almost surely,
1
O
sup sup tPCn ≤ n
t∈0 T C∈
for all n > N (6.7)
O
Because PC ⊂ PCn for all C ∈ and n ∈ , conclude that almost surely,
sup sup tPC = 0
t∈0 T C∈
We now establish property (ii). Recall that Zt = 1 t for all t ≥ 0, and Su v =
v ≥ u ≥ 0.
v
u
1/Zs ds for all
Theorem 6.3. Almost surely, the process · satisfies
tA = 0A + S0 t t + for all t ≥ 0 and A ∈ .
0
t
A + Ss t t − s ds
(6.8)
Gromoll, Robert, and Zwart: Fluid Limits for Processor-Sharing Queues with Impatience
399
Mathematics of Operations Research 33(2), pp. 375–402, © 2008 INFORMS
Proof. Let T > 0. It suffices to show that almost surely, (6.8) holds for all t ∈ 0 T and all A ∈ . For
each r ∈ , define the random variable
r s tC − r t − s˘ r C
XTr = sup sup (6.9)
C∈ 0≤s≤t≤T
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Pq
By Lemma 5.1, XTq → 0 as q → . Because the limit is deterministic, this convergence is joint with the
q · ⇒ ·. Using the Skorohod representation theorem, assume without loss of generality that
convergence q
q
· XT and · are defined on a common probability space such that
q · XTq → · 0 almost surely.
(6.10)
The conclusions of Lemmas 6.1 and 6.2 hold almost surely as well. Assume for the remainder of the proof that
all random objects are evaluated on the event of probability one such that · is continuous and such that (6.1),
(6.5), and (6.10) hold.
Fix t ∈ 0 T and C ∈ . An extension to all Borel sets A ∈ will be made at the end. For each q, (5.10)
yields
q
1 q Et + q
q
q
q
iq t
tC = 0C + S 0 t t +
1 B t D
(6.11)
q i=1 C i
We will obtain (6.8) from (6.11) by letting q → . The convergence in the first component of (6.10) is in the
Skorohod J1 -topology on D0 M. However, because · is continuous,
w
q s →
s
for all s ∈ 0 t.
q · and Z· = 1 ·, this implies that
Because Z q · = 1 lim Z q · − Z· = 0
t
q→
(6.12)
(6.13)
For all t ≥ v ≥ u > 0, (6.1) implies that inf s∈uv Zs > 0, and so the bounded convergence theorem yields
v 1
ds
lim Sq u v = lim
q s
q→
q→ u Z
v 1
=
ds
u Zs
= Su v
(6.14)
If Z0 $= 0, then (6.14) holds for u = 0 as well, because then inf s∈0v Zs > 0. If Z0 = 0, then S0 v = and Sq 0 v → as q → .
Suppose that Z0 $= 0 and let 9 > 0. By (6.14), there exists a q9 ∈ such that Sq 0 t ∈
t − 9+ S0
t + 9 for q > q9 . Deduce from the shape of the set C, (6.12), and (6.5) that
S0
q 0C + Sq 0 t t ≤ 0C
t − 9+ t
+ S0
lim sup q→
q 0C + Sq 0 t t ≥ 0C
t + 9 t
lim inf + S0
q→
By (6.5), letting 9 → 0 yields
q 0C + Sq 0 t t = 0C
t t
+ S0
lim q→
(6.15)
q 0 = 0 by (6.13).
If Z0 = 0, then (6.15) holds trivially because the left side is bounded above by limq→ 1 q
Combining with (6.12) and (6.5) for t implies that, as q → ,
q 0C + Sq 0 t t → tC − 0C + S0 t t
q tC − Let I q denote the second right-hand term in (6.11). Let # > 0 and let K ∈ 0 t. Because Sq s t is nonincreasing in s and S· t is continuous on K t, (6.14) implies that Sq · t → S· t uniformly on K t. That
is, there exists q# ∈ such that
sup Sq s t − Ss t ≤ #
s∈Kt
for all q > q# (6.16)
Gromoll, Robert, and Zwart: Fluid Limits for Processor-Sharing Queues with Impatience
400
Mathematics of Operations Research 33(2), pp. 375–402, © 2008 INFORMS
+ and + × · are
Let D = C ∈ " PC $= 0. Note that D is countable because · × probability measures. Because Zu > 0 for all u ∈ K t, the function Ss t is strictly decreasing in s on K t.
Thus,
D S = s ∈ K t" C + Ss t ± 2# t − s ∈ D INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s).
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is also countable. For each integer N > 1, let K = t0N < t1N < · · · < tNN = t be a partition of K t such that
N
− tjN → 0 as N → . Then,
tjN ( D S for all j = 1 / / / N − 1, and such that maxj≤N −1 tj+1
q Eq t N N
−1
j+1
1
1 q EK + q
iq t +
q
q
I =
1C Bi t D
1+
C Bi t Di t
q i=1
q
N
j=0
i=q Eq t +1
q
q
j
N
q 0 K
2+ . Suppose that tjN ≤ Uiq q −1 ≤ tj+1
, for
Note that the first right-hand term is bounded above by q
q
some q > q# , some j ≤ N − 1, and some i ∈ q E K + 1 / / / q E t. Then, by (6.16),
N
Stj+1
t − # ≤ Sq Uiq q −1 t ≤ StjN t + #
By definition,
(6.17)
iq t = Biq − Sq Uiq q −1 t Diq q −1 − t − Uiq q −1 Biq t D
So, for q > q# , (6.17) and the inequalities 1C · − # · ≤ 1+
C · · ≤ 1C · + # · yield
q
q −1
N
q
q
1+
− t − tjN 3
C Bi t Di t ≥ 1C Bi − Stj t + 2# Di q
q
q −1
N
N
q
q
1+
− t − tj+1
C Bi t Di t ≤ 1C Bi − Stj+1 t − 2# Di q
This yields, for q > q# ,
q
I ≥
N
−1
j=0
q
q
I ≤
q Eq t N j+1
1
1 B q − StjN t + 2# Diq q −1 − t − tjN 3
q i=q Eq tN +1 C i
j
N
−1
2+ +
0 K
j=0
q Eq t N j+1
1
N
N
1 B q − Stj+1
t − 2# Diq q −1 − t − tj+1
q i=q Eq tN +1 C i
j
Rewrite as
Iq ≥
N
−1
j=0
q
q
I ≤
N
q tjN tj+1
C + StjN t + 2# t − tjN 3
2+ +
0 K
N
−1
j=0
(6.18)
q
N
tjN tj+1
C
N
+ Stj+1
t − 2# t
N
− tj+1
By (6.9) and (6.18), q > q# implies that
Iq ≥
N
−1
j=0
N
q tj+1
− tjN ˘ q C + StjN t + 2# t − tjN − XTq 3
I q ≤ q K + XTq +
N
−1
j=0
N
N
N
q tj+1
− tjN ˘ q C + Stj+1
t − 2# t − tj+1
+ XTq By (6.10) and because tjN $∈ D S for all j = 1 / / / N − 1,
lim inf I q ≥ q→
N
−1
j=0
N
tj+1
− tjN C + StjN t + 2# t − tjN 3
(6.19)
lim sup I q ≤ K + q→
N
−1
j=0
N
N
N
tj+1
− tjN C + Stj+1
t − 2# t − tj+1
Gromoll, Robert, and Zwart: Fluid Limits for Processor-Sharing Queues with Impatience
Mathematics of Operations Research 33(2), pp. 375–402, © 2008 INFORMS
401
For s ∈ K t such that s ( D S, the bounded convergence theorem implies that
lim
N
−1
N →
lim
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N →
j=0
N
−1
j=0
N
N
N sC + St t + 2# t − t = C + Ss t + 2# t − s3
1tjN tj+1
j
j
(6.20)
N sC
1tjN tj+1
N
+ Stj+1
t − 2# t
N
− tj+1
= C
+ Ss t − 2# t − s
Thus, the convergence in (6.20) holds for almost every s ∈ K t. Let N → in (6.19) and conclude from
(6.20) and the bounded convergence theorem that
t
lim inf I q ≥ C + Ss t + 2# t − s ds3
q→
K
lim sup I q ≤ K + t
K
q→
C + Ss t − 2# t − s ds
(6.21)
Let # → 0 in (6.21). Because D is countable, both integrands in (6.21) converge almost everywhere on
K t to C + Ss t t − s. Thus,
t
lim inf I q ≥ C + Ss t t − s ds3
q→
K
lim sup I q ≤ K + q→
Let K → 0 to conclude that
lim I q = q→
t
0
K
t
C + Ss t t − s ds
C + Ss t t − s ds
This proves (6.8) for all t ∈ 0 T and C ∈ . To extend to all A ∈ , apply the 4-argument appearing
in §2.3. References
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